The measurement problem is a problem in Quantum Mechanics. Although its decribed as a philosophical problem in Physics - it is in fact a problem of physics. This is how Bohr, Einstein & Heisenberg when discussing this would have seen it. It is dscribed as such now as the mathematics of QM is very well understood now - so it has become a seperate question on the interpretation of QM.
Its worth bearing in mind, though, an older question in the philosophy of physics, to see the physical relevance of so-called philosophical questions. One of the criticisms faced by Newtons very successful theory of Universal Gravitation, universal because it tied together the separate domain of terrestial & celestial motion (in Aristotles theory of Physics) under the rubric of one force, is the question of force acting at a distance. How does force that originate at the Sum affect the motion of the planet Earth here? It took three centuries before that particular question was solved when Einstein realised it was the very fabric of space-time that transmitted the force.
One might say a problem of physics becomes philosophical when it is a question too diffcult to answer with the mathematical & physical technology then at hand.
Now, Quantum Mechanics describes a physical system by a probability wave which evolves in two distinct ways. It evolves in a deterministic manner until a physical property of the system is measured and then it evolves in a non-determinstic manner and collapses into a value that is measured by the observer.
Everyone agrees on what QM says, the problem comes with interpreting the situation described so precisely by the mathematics/physics of QM.
Bohr, in what is known as the Copenhagen interpretation, stated that the measurement of the physical system was dependent in a crucial way on the experimental situation including that of the observer. One could say that it is observer dependent. From a philosophical point of view this has a certain family resemblence with Kantian Metaphysics.
Heisenberg took a different slant on the same problem - he said that the physical system had no physical properties as such until it was measured. That is his metaphysics was positivist or Instrumentalist.
Einstein himself took a realist position on the measurement problem - he insisted that the physical system had physical properties at all times. This is in fact the classical point of view, as one would be familiar with from Newtonian Mechanics.
So here we have three philosophical interpretations - Neo-Kantian, Positivist & Realist. The realist position being traditional in Classical physics. Positivist since the development of 'positive' science in Vienna at the turn of the 20C, when one could say the very basis of science turns itself into a philosophy - nothing is unless it can be measured by some instrument. Neo-Kantian takes off from the Kantian position where the observer conditions how experience is to be made intelligible.
Notably several questions are left unclarified by the usual treatments - why the separation between a macroscopic realm where observations are done by observers that are classical, and the observed Quantum Mechanical realm. Secondly must observers be concious? Third - why is there a deterministic phase followed by a non-determinstic collapse?
There are certain results of QM (Bells Inequalities, Kockhen-Specker Theorem) that show that a realist local theory is not possible. That is any realist interpretation of QM must have non-local (faster than light propagation) correlations (as in the Einstein-Podolsky-Rosen Gedanken Experiment). In fact such a theory has been constructed by Bohm - de Broglies pilot wave theory - otherwise known as Bohmian Mechanics. Of course it was only with the advent of Einsteins theory of relativity that the speed of light was understood as a fundamental limit, a violation of it would indicate a violation of causality.
Two modern solutions of the interpretational problem of QM is Relational Quantum Mechanics as put forward by Rovelli, and Consistent Histories by Isham and others.
In Relational QM, one democratises the idea of an observer. First, if I observe a system then that system is observing me. Secondly, an observer may as well be an electron as well as some human being. Conciousness is not the marker of an observer rather it is the ability to be affected by what is observed. So in a manner of speaking, a planet observes the gravitational field of the sun. An electron observes the electro-magnetic field, and conversely the electromagnetic field observes the electron. Here, 'observe' is a synonym for to affect or be affected by. In this view, there are no special interactions that we choose to call measurements - all interactions are measurements.
Consistent Histories is used with Quantum Decoherence to establish which sequence of propositions can be meaningfully asked of a system. The decoherence of the quantum mechanical system in its environment is used to statistically 'magick' away the wave-packet collapse. One argument against decoherence is that one should include decoherence with the physical system under observation, but given the fluctuations one expects at the planck-scale shows that conceptually that this is not practicable.
There is of course Everetts 'multiverse' of parallel universes, which offers the benefit of a realist interpretation at the enormous expense of an enormous multiplication of universes, but I find this enormous 'bristley' multiplication of universes somehow unflattering to QM, and requires a shave by Occams Razor...